Upcoming Event: Oden Institute Seminar

Gradient-based dimension reduction + measure transport for Bayesian inverse problems

Dr. Michael Brennan, MIT

Abstract

Inference is a pervasive task in science and engineering applications. The Bayesian approach to inference facilitates informed decision-making by quantifying uncertainty in parameters and predictions but can be computationally demanding. In the case of Bayesian methods for inverse problems governed by partial differential equations (PDEs), the high dimensionality of model parameters and data can render naïve posterior exploration intractable.

One strategy for reducing the computational cost of inference in these settings is to discover and exploit some type of low-dimensional structure in the posterior. In this talk, we present methods for discovering two different notions of low-dimensional structure using gradients of the unnormalized posterior and parameter-data joint log-densities. We then link these dimension reduction methods to inference algorithms that employ measure transport, where one expresses the target posterior distribution as a transformation of a simple reference distribution (e.g., a standard Gaussian).These methods substantially decrease the computational burden of accurate inference in high-dimensional problems and reveal interpretable structure that provides qualitative insights. We demonstrate the benefits of each method on Bayesian inverse problems related to steady-state Darcy flow and inverse scattering.

Biography

Dr. Brennan is a postdoctoral associate at MIT working with Prof. Youssef Marzouk of the Uncertainty Quantification Group. He is broadly interested in numerical methods for uncertainty quantification and probabilistic modeling. Currently he works on techniques that discover and exploit low-dimensional structure to accelerate statistical inference.

He received his PhD at MIT in Computational Science and Engineering in 2023. His thesis focused on dimension reduction for Bayesian inverse problems and simulation-based inference. He received his Master’s degree in Mathematics from Virginia Tech, where he studied nonlinear eigenvalue problems and reduced order modeling.